A characterization of definability by positive first order formulas in terms of Fraiss'e-Ehrenfeucht-like games is developed. Using this characterization, an elementary, purely combinatorial, proof of the failure of Lyndon's Lemma (that every monotone first order property is expressible positively) for finite models is given. The proof implies that first order logic is a bad candidate to the role of uniform version of positive Boolean circuits of constant depth and polynomial size. Although Lyndon's Lemma fails for finite models, some similar characterization may be established for finitely monotone properties, and we formulate a particular open problem in this direction. 1 Introduction In the Winter of 92/93, I taught a graduate course in Finite Model Theory at the Mathematics Department of UCLA. Although the experience was very satisfying, at least for me, I continued to feel somewhat unsatisfied about a few things, among them the most important for me was the result by Ajtai and G...

. Logic preservation theorems often have the form of a syntax /semantics correspondence. For example, the / Los-Tarski theorem asserts that a first-order sentence is preserved by extensions if and only if it is equivalent to an existential sentence. Many of these correspondences break when one restricts attention to finite models. In such a case, one may attempt to find a new semantical characterization of the old syntactical property or a new syntactical characterization of the old semantical property. The goal of this paper is to provoke such a study. 1 Introduction It is well known that famous theorems about first-order logic fail in the case when only finite structures are allowed (see, for example, [?]). A more careful examination shows that it is wrong to lump all these failing theorems together. On one side we have theorems like completeness or compactness where the failure is really and truly hopeless. On the other side there are theorems like the / LosTarski theorem, which we...

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Finite Model Theory and Finite Variable Logics Eric Rosen Supervisor: Scott Weinstein In this dissertation, I investigate some questions about the model theory of finite structures. One goal is to better understand the expressive power of various logical languages, including first-order logic (FO), over this class. A second, related, goal is to determine which results from classical model theory remain true when relativized to the class, F , of finite structures. As it is well-known that many such results become false, I also consider certain weakened generalizations of classical results. I prove some basic results about the languages L k (9) and L k 1! (9), the existential fragments of the finite variable logics L k and L k 1! . I show that there are finite models whose L k (9)-theories are not finitely axiomatizable. I also establish the optimality of a normal form for L k 1! (9), and separate certain fragments of this logic. I introduce a notion of a `generalized preser...

In this paper we study the lower bounds problem for monotone circuits. The main goal is to extend and simplify the well known method of approximations proposed by A. Razborov in 1985. The main result is the following combinatorial criterion for the monotone circuit complexity: a monotone Boolean function f(X) of n variables X = fx 1 ; : : : ; x n g requires monotone circuits of size exp(\Omega\Gamma t= log t)) if there is a family F ` 2 X such that: (i) each set in F is either a minterm or a maxterm of f; and (ii) D k (F)=D k+1 (F) t for every k = 0; 1; : : : ; t \Gamma 1: Here D k (F) is the k-th degree of F , i.e. maximum cardinality of a subfamily H ` F with j " Hj k: 1 Introduction The question of determining how much economy the universal non-monotone basis f; ; :g provides over the monotone basis f; g has been a long standing open problem in Boolean circuit complexity. In 1985, Razborov [10, 11] achieved a major development in this direction. He worked out the, so-called,...

A class of structures is said to have the homomorphism-preservation property just in case every firstorder formula that is preserved by homomorphisms on this class is equivalent to an existential-positive formula. It is known by a result of Rossman that the class of finite structures has this property and by previous work of Atserias et al. that various of its subclasses do. We extend the latter results by introducing the notion of a quasi-wide class and showing that any quasi-wide class that is closed under taking substructures and disjoint unions has the homomorphism-preservation property. We show, in particular, that classes of structures of bounded expansion and that locally exclude minors are quasiwide. We also construct an example of a class of finite structures which is closed under substructures and disjoint unions but does not admit the homomorphism-preservation property.

The homomorphism preservation theorem (h.p.t.), a result in classical model theory, states that a first-order formula is preserved under homomorphisms on all structures (finite and infinite) if and only if it is equivalent to an existential-positive formula. Answering a long-standing question in finite model theory, we prove that the h.p.t. remains valid when restricted to finite structures (unlike many other classical preservation theorems, including the ̷Lo´s-Tarski theorem and Lyndon’s positivity theorem). Applications of this result extend to constraint satisfaction problems and to database theory via a correspondence between existential-positive formulas and unions of conjunctive queries. A further result of this article strengthens the classical h.p.t.: we show that a first-order formula is preserved under homomorphisms on all structures if and only if it is equivalent to an existential-positive formula of equal quantifier-rank.

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We prove that the homomorphism preservation theorem (h.p.t.), a classical result of mathematical logic, holds when restricted to finite structures. That is, a first-order formula is preserved under homomorphisms on finite structures if, and only if, it is equivalent in the finite to an existential-positive formula. This result, which contrasts with the known failure of other classical preservation theorems on finite structures, answers a longstanding question in finite model theory. The relevance of this result, however, extends beyond logic to areas of computer science, including constraint satisfaction problems and database theory; the database connection arises from a correspondence between existential-positive formulas and unions of conjunctive queries (also known as select-project-join-union queries). A second result of this article strengthens the classical h.p.t. by showing that a firstorder formula is preserved under homomorphisms on all structures if, and only if, it is equivalent to an existential-positive formula of equal quantifier-rank. Unlike traditional proofs of the classical h.p.t., the proof of this stronger “equirank ” theorem is compactnessfree and constructive. While these results are logical in nature, the technical development of the article takes place almost entirely within a combinatorial framework. The concept of tree-depth, a graph parameter related to tree-width, plays an important role in our analysis (as a combinatorial counterpart to quantifier-rank). We introduce new notions of n-homomorphism and n-core, which approximate the familiar concepts of homomorphism and core “up to tree-depth n”. The key technical lemmas take a pair of n-homomorphically equivalent [finite] relational structures and construct corresponding [finite] co-retracts which satisfy a certain back-andforth property.

A set A of vertices of a graph G is called d-scattered in G if no two d-neighborhoods of (distinct) vertices of A intersect. In other words, A is d-scattered if no two distinct vertices of A have distance at most 2d. This notion was isolated in the context of finite model theory by Gurevich and recently it played a prominent role in the study of homomorphism preservation theorems for special classes of structures (such as minor closed families). This in turn led to the notions of wide, almost wide and quasi-wide classes of graphs. It has been proved previously that minor closed classes and classes of graphs with locally forbidden minors are examples of such classes and thus (relativized) homomorphism preservation theorem holds for them. In this paper we show that (more general) classes with bounded expansion and (newly defined) classes with bounded local expansion and even (very general) nowhere dense classes are quasi wide. This not only strictly generalizes the previous results but it also provides new proofs and algorithms for some of the old results. It appears that bounded expansion and nowhere dense classes are perhaps a proper setting for investigation of wide-type classes as in several instances we obtain a structural characterization. This also puts classes of bounded expansion in the new context. Our motivation stems from finite dualities. As a corollary we obtain that any homomorphism closed first order definable property restricted to a bounded expansion class is a restricted duality.